Problem — Conservation of energy with friction

Conservation and friction. A block slides down an inclined plane with friction. Explain how the total energy is conserved, indicating where the heat produced goes.

Solution

The block starts at height $h$ and descends along the plane over a stretch $L$. Without friction all the potential energy would become kinetic ($mgh = \tfrac12 mv_f^2$). Kinetic friction, force $f = \mu_d N = \mu_d mg\cos\theta$, does negative work over the stretch $L$ and removes the energy

$Q = f\,L = \mu_d\,mg\cos\theta\;L$

The complete energy balance at the base of the plane is therefore:

$\ev{mgh = \tfrac{1}{2}mv_f^2 + Q}$

Where the heat goes. The total energy is always conserved: nothing disappears. What friction does is transform ordered macroscopic kinetic energy (the block descending as a whole) into disordered microscopic kinetic energy: the molecules at the surface of the block, the plane and the surrounding air vibrate more intensely. This is the heat $Q$: a slightly higher temperature of block and plane in the contact zone. Conservation holds, but that share of energy becomes inaccessible — it cannot be fully recovered as ordered motion (this is thermodynamics’ arrow of time).

Collegamenti

Argomenti: Lavoro ed energia Concetti: Energia termica da attrito · Conservazione dell’energia meccanica Competenze: Scomposizione sul piano inclinato Oggetti: Piano inclinato · Blocco